Our arguments in both of these cases mirror those of [1]}, with (REF ) and (REF ) in place of Lemmas 3.6 and 3.7. The arguments from [1]} show that in the dense case, we must have either
\(K_2K_3^8K_4^4 \gtrapprox \delta ^{\alpha }s^{-1}\quad \textrm {and/or}\quad K_1^3K_2K_3^8K_4^4 \gtrapprox \delta ^{-\gamma \alpha },\) 
